A. Appendices

A.2 Software Resources

I amassed a large collection of software available for Mac OS computers from 1985~2005.
Most of these are shareware or freeware. Some were used to create the graphics included
in this book. Others were used for inspiration or entertainment.
Files archived on this server were compressed using
StuffIt 5.5.

OS X

Dynamical Systems

1D MapsJames D. Meiss, 2001James D. MeissSeven different folded mappings (logistic, cosine, cubic, exponential, tangent, tent, and circle map). Five different ways to analyze them (cobweb, bifurcation, lyapunov exponent, invariant density, and exit time plot). Interesting in a limited sort of way. Also available in a version for the Classic OS.

ABC mapJames D. Meiss, 2003James D. MeissA 3D map editor. An example of what I would call an "unripened" application. As the author himself warns, "This is NOT a full blown nice Mac application… and using this will possibly crash the program if you type in the wrong numbers. USE AT YOUR OWN RISK!" I concur.

Chaotic FlowsJohn Lindner, Bryan Prusha, Josh Bozeday, 2001CoWCPPExplore the Chua, Duffing, Lorenz, and Rössler attractors. Zoom in and out, rotate, and play with the parameters. Also available in a version for the Classic OS. One of many scientific applications for the Macintosh from the very interesting College of Wooster Computational Physics Project. Other applications at this site related to chaos and fractals include:

C-Ball: Interactively explore the chaotic dynamics of a ball bouncing between two circles.

Gerry's Attraction (formerly Gerry's Lorenz Attractor) Gerry Beggs, 2004Gerry BeggsStart with a group of points on the xy plane, run them through the right equations, and watch them congeal on the Lorenz Attractor or the less interesting Rössler Attractor. Also comes with an option that allows you to fly around the attractors while they evolve. An inspired way to view them. Much better than fixed diagrams. Carbonized to run in both the Classic and OS X environments.

PopulusDon Alstad, 2003Don AlstadA comprehensive tutorial on population and evolutionary dynamics with something like 20 different models to play with. The help files are more interesting than the program itself, however, which hesitates in the most annoying manner. Still has a lot of potential as a great teaching tool for undergraduate students in this field. Also available in a version for Windows, Linux, Unix, DOS and Java. (The OS X version is actually just a "packaged" Java application.)

StageGrowIan G. Gillespie, 2001Ian GillespieA program for exploring population dynamics in a single species. Very technical. Designed for people who know what it's all about (which excludes me). Staged, geometric, exponential, and logistic models are supported.

StdMapJames D. Meiss, 2002James D. MeissStandard map orbits in this book were drawn using the Classic version of this program. Has other features for exploring nine different area preserving maps. A nice piece of software with good documentation. Fast and fun to play with, too.

Lindenmayer Systems

Fractal GrowerJoel Castellanos, 2007Joel CastellanosJava software for growing L-systems fractals. Being Java means it works on multiple platforms, including the browser you're using to view this web page. Make your own system or use one of the many presets (paper folding, bush, carpet, dragon, fern, big-h, twig, weed, koch snowflake, sierpinski triangle, etc.).

FractalTrees X (also known as FTX)Simon Woodside, 2002Simon WoodsideDraws simple stick figure, fractal trees. Comes with an easy to operate set of sliders for changing parameter values. Orders of magnitude better than the original Classic application. The author is also working on a screen saver version called FTXSaver.

VonKochJulien Salort, 2003Julien SalortA simple, little program for drawing the Koch snowflake and its variants.

Mandelbrot & Julia Sets

AltiVec Fractal CarbonDauger Research, 2001-2004Dauger ResearchA program for testing the computing power of a distributed computing network. Generates images of the standard and quartic Mandelbrot sets, then tells you the speed of your parallel processor network in megaflops. If you only have one computer, then it tells you the speed of just that computer. Allows you to zoom in and that's about all. Comes with very few options as it is primarily a benchmarking tool. Also available as a Classic application with no options — zero. It runs and it's done.

Carbon FractalHarold Cooper, 2001physoft [last url]carbon-fractal.sitIt appears as if it just draws Mandelbrot and Julia sets, but the website claims it has additional features. It doesn't really matter as it is too slow to be much fun. Simple programs like this one should run fast and be intuitively obvious to operate. This one is neither.

Cheap MandelConor McCarthy, 2002Conor McCarthyExamine the Mandelbrot set. A beginner's attempt at programming for the Mac. The original Classic OS version was not much better.

Complex ExplorerThomas Hallock, 2002Thomas HallockA very simple program for exploring the Mandelbrot set. Comes with a unique color palette editor that I really like. Allows the user to save coordinates, but not images. A version for the Classic OS apparently exists, but I've never tested it.

DynamicsRichard Koch, Pierre d'Herbemont, 2001Richard KochA very basic fractal set explorer. Zoom and pan around a colored Mandelbrot set in the main window. View the associated black and white Julia set in a side window. A bit slow, given all the speed claims for OS X native applications.

EasyFractalBerkhan Software, 2004Berkhan SoftwareZoom into the Mandelbrot set, check out the corresponding Julia set, play with the colors, try a different function. Comes with a collection of preset parameter files, some of which are interesting and rather unusual. Available in German and English versions for both the Classic OS and OS X.

EDFractalEdward Baskerville, ED Development, 2001ED Development [last url]edfractal.sitThe first chaos/fractal application written for OS X that I found. Explore the Mandelbrot set. Create Julia sets and explore them, too. A very straightforward example of programming.

FFFF - Fast Floating Fractal FunDaniele Paccaloni, 2005Source Forge Developer ProfileThe crude beginings of an application that draws the Mandelbrot set, allows you to zoom in (but not out), rotate the color scheme and a bit more. Runs in terminal mode on OS X. Very speedy. Also available in a Windows version.

Fractal Domains (formerly FracPPC)Dennis C. De Mars, 1994-2004Fractal DomainsExplore the Mandelbrot set. Switch to the accompanying Julia set. Zooming in and out is rather tiresome, but otherwise this is a fine application. The switch from the Classic version to OS X has brought about a much improved look and feel.

Fractale de Julia, Fractale de Mandelbrojt (a.k.a. Fracatle Julia, Fracatle Mandel)Didier Straus, 2002Software-DSStudent programming projects that don't work. Draws a filled Julia or Mandelbrot set, then quits (or never completes the startup process). These application should not have been made public. They are effectively locked in an alpha phase of arrested development.

GastonLeo Fink, 2003gaston.sitA nice little program that renders three dimensional cross sections of quaternion Julia sets. A quaternion is a type of complex number with four parts (r, i, j, and k) rather than two (real and imaginary). Adjust all four parts of the quaternion constant with sliders. Try different escape radii until you find an interesting looking surface and export it as a TIFF. Also has sliders for adjusting the precision and resolution. Given the complexity of quaternion mathematics, you will want to keep the resolution low during casual exploration. High quality images are rendered very slowly. My only complaint is that the sliders are somewhat crude and hard to adjust finely. There should be some method for enetring the constant values directly.

Mandel BlotNixanz, 2002nixanz.comAmateur night at the Mac. Draw and explore the Mandelbrot set. Drag to zoom in, poof, the image window disappears, poof, the image window reappears with your new image. Very simple color options: your choice of red, green, blue, cyan, magenta, or yellow with varying brightness and contrast. Note how the author has chosen to hide behind a screen name.

Mandel BotRobert Covington, 2003Artly ThereAccording to the author, "A simple Mandelbrot viewer based on a demo by Robert Purves." Quite right.

MandelbrotChristoph Lauer, 2002Christoph LauerA painfully slow Java program for exploring the Mandelbrot set and rendering corresponding Julia sets. Java is not the right environment for applications that require intensive calculation. Also available for Linux and Windows.

MandelbrotDavid Ross, 2001David RossAn easily forgettable program. Draw and explore Mandelbrot sets from order 2 through 59. This may sound impressive but really isn't. Very high order Mandelbrot sets all look like circles. Switch over to the Julia set associated with a particular location of the Mandelbrot set. Very simple and plain color schemes.

Mandelbrot Viewer (a.k.a. Custom View)Michael C Thornburgh, 2001Mr. MikeA good basic Mandelbrot explorer. Simple and fast with very few options. Zoom in, but not out. Adjust the escape radius, number of iterations, number of threads (not quite sure what that means), and location of the origin (which renders distorted Mandelbrot-Julia hybrids), but not the color scheme.

mandelPathAaron Golden, 2003Aaron GoldenDoesn't draw the Mandelbrot set in the traditional manner (counting the number of iterations to escape). Instead it traces the orbit of points in the complex plane. As the author says, "It's not particularly useful but I think the pictures are pretty." This program has to be run through a terminal window. Very odd.

PhoenIXAdrian Platts, 2001Adrian PlattsDeliberately styled interface with options, options, options. Auto zooms, generates relief maps, makes movies, plays background MIDI and MP3 files, and acts as a screen saver. It also draws several kinds of Mandelbrot and Julia sets. Comes with yet more options that I haven't described. Also available in a version for the Classic OS.

TransWorld (a.k.a. WindowFreak)Junzo Sato, 2001Junzo SatoA waste of time. Very slowly renders a very low resolution image of the full Mandelbrot set or one specific Julia set in a transparent window. No zooming. No variations of any sort other than the transparency level (thus the name TransWorld). A "look what I can do" application that is of interest to no one.

Newton's Method

NewtonFractalStefan Messmer, 2001Stefan MessmerDraw fractals using Newton's root approximation method in the complex plane. Faster and more reliable than the original Classic application, but still looks and acts like a beta edition in need of a solid debugging. Also available in Windows and Java versions.

Terrain & Landscape Modeling

Cosmos3dJeff Biggus & Heinz Nabielek, 2004osx.hyperjeff.netA program whose sole purpose is to render Hoyle's galaxy clumping model as mentioned on page 218 of Mandelbrot's The Fractal Geometry of Nature. A good fractal for those interested in the history of cosmology, but I think Hoyle's model was disproved by galactic surveys.

MojoWorld Generator DemoKen Musgrave, Pandromeda Inc, 2002PandromedaA surrealistic (super-realistic?) landscape generator. A sophisticated, commercial application that is not for the casual user. The gallery of images on the company's website hints at the possibilities for someone willing to put in the time and effort: mountain ranges, beaches, rivers, clouds, sand dunes, moons, entire planets. Also available in a version for Windows.

Miscellaneous

CellularChristian Grunenberg, 2001Christian GrunenbergA program essentially designed to investigate cellular automata that also happens to draw Mandelbrot sets and model galactic collisions. Not bad, but nothing special.

Escape (a.k.a. EscapePPC)Graham Anderson, 2004EscapePPCA comprehensive application for exploring over 20 different types of fractal images and strange attractors. Unregistered versions can only render Mandelbrot sets, however. Many, many features to explore for those with a serious interest. Carbonized to run in both the Classic and OS X environments. (Not to be confused with EscapeFractals: a Newton's Method application for the Classic OS.)

Gerry's Mandelbrot SetGerry Beggs, 1998Gerry BeggsMore than just the Mandelbrot set, it also creates Julia sets, IFS images, a fractal tree, the Lorenz attractor, bifurcation diagrams, Newton's method, and Martin's mapping (a.k.a. hopalong). Gives you the ability to check out a lot of different fractals using just one application. Very fast. Also available in a version for the Classic OS.

Screen Savers

Apfelmaennle, HuepferHeiko Kretschmer, 2002wurst-wasser.netApfelmaennle draws the whole Mandelbrot set, selects a region, and then magnifies it. Selects a subregion and magnifies that. Keeps going and going. I've never run it long enough to see it run out of steps, but eventually it must. According to the author, Huepfer is "One more screen saver displaying fractal-like graphics known as Hopper or Hüpfer." He does not explain what this is, however, nor have I ever heard of it before. Sometimes it looks like a diffusion pattern, sometimes like a checkerboard. Apfelmaennle is the better of the two programs.

AmseqAndré and Paul van der Kouwe and Thomas Benner, 2003Thomas BennerAmseq stands for "Animated Mandelbrot Sequence Generator". There is no documentation to explain what it does, but it appears to trace the orbit of points in the complex plane through successive iterations of the quadratic mapping. Recall that the Mandelbrot set is the set of all points that don't escape to infinity after an infinite number of iterations of the quadratic mapping. This screen saver draws the path of the points used to draw the Mandelbrot set, not the set itself.

Coral, Flame, Hopalong, ifsTommaso Pecorella, 2001, 2002UselesssoftOpen source Unix programs written in the late '80s and early '90s transformed into screen savers for OS X. Coral fills the screen with DLA (diffusion limited aggregation) fractals that look something like corals when shaded in. Flame draws "weird cosmic fractals" (whatever those are). Hopalong draws the scribbly fractal of the same name plus Martin's Method, Renaldo Recuerdo (called RR), and EJK fractals (whatever those are). ifs draws an iterated function system that evolves smoothly over time. The latest addition to this group, ifs is by far the best. Using Hopalong to save your screen seems risky as it renders images that stay static on the screen for long periods.

FTXSaverSimon Woodside, 2004Simon WoodsideFTX means FractalTrees X — the name of an application from which the screen saver was derived. Draws simple stick figure, fractal trees that evolve on the screen. Still in the alpha phase but has potential. Looks like it will make interesting images, but is just too computationally rough in its current form.

HénonMarco Coïsson, 2003Marco CoïssonThis screen saver gives a graphical representation of the so-called Hénon correlations. Simple and uneventful. Available in English and Italian versions.

IFS3DDavid Leppik, 2001David LeppikFlies over and around an evolving IFS fractal, but I don't like the way it looks. The animation isn't smooth and the images are unnaturally dim.

Sierpinski 3DEpicware, 2001Epicwarescreen saver of a flying, tumbling Sierpinski pyramid. Part of an Open GL bundle of OS X screen savers.

XeverywhereStéphane Sudre, 2000-2002White BoxStart with a cube, replaced it with smaller cubes, replace the smaller cubes with even smaller cubes, and so on, then work backward. Comes with a warning not to set the number of iterations too high. Take it seriously or the Xeverywhere will seize control of your CPU.

Classic OS

Dynamical Systems

1D MapsJames D. Meiss, 2001James D. MeissSeven different folded mappings (logistic, cosine, cubic, exponential, tangent, tent, and circle map). Five different ways to analyze them (cobweb, bifurcation, lyapunov exponent, invariant density, and exit time plot). Interesting in a limited sort of way. Also available in a version for OS X.

1-D Chaos ExplorerMatthew Hall, 19921-d-chaos-explorer.sitBifurcation diagrams, web diagrams, time series, etc. Program your own functions for exploration. This is a good program for those who want to understand the basic behavior of iterated systems. Many of the web and bifurcation diagrams in this book were rendered with this program.

BifurcationRonald T. Kneusel, 1995bifurcation.sitFaster than 1‑D Chaos Explorer, but does half as much (bifurcation, and time-series only).

Bouncing BallT. Abbott, N. B. Tufillaro, J. P. Reilly, 1993bouncing-ball.sitDo you find the quadratic map too abstract and the logistic function an oversimplification? Steady state, periodic, and chaotic phase space orbits can all be illustrated by a ball bouncing on a vertically oscillating table. It's a great physical application, but I find the program hard to work with and slow. There are too many windows and too many options. It will take some effort to master all of them, so beware. One of the authors has a page describing the program, but it is no longer available for download. This program and Quadratic Map (see below) go with the book An Experimental Approach to Nonlinear Dynamics and Chaos, the entire contents of which are online in html or pdf format.

ChaosPlotJason Regier, 1994chaosplot.sitOrbit diagrams for continuous systems. Plots the behavior of a damped, driven, anharmonic oscillator. Generates a chaotic path reminiscent of the shadow of a fly on a wall.

Chaotic FlowsJohn Lindner, Bryan Prusha, Josh Bozeday, 1997CoWCPPExplore the Chua, Duffing, Lorenz, and Rössler attractors. Zoom in and out, rotate, and play with the parameters. Also available in a version for OS X. One of many scientific applications for the Macintosh from the very interesting College of Wooster Computational Physics Project. Other applications at this site related to chaos and fractals include:

Celestial Chaos: Illustrates chaos in the restricted 3-body problems by simulating an asteroid moving in the gravitational field of a binary star system.

Cycle ExplorerJames C. Burgess, 1995Jim BurgessClick on the bifurcation diagram and draw the corresponding web diagram. Move the parabola around on the web diagram and see the corresponding location on the bifurcation diagram. Very limited interface with a wise guy attitude. Pulling down the "Options" menu gets "No Options" as a reply.

Gerry's Attraction (formerly Gerry's Lorenz Attractor)Gerry Beggs, 2002Gerry BeggsStart with a group of points on the xy plane, run them through the right equations, and watch them congeal on the Lorenz Attractor or the less interesting Rössler Attractor. Also comes with an option that allows you to fly around the attractors while they evolve. An inspired way to view them. Much better than fixed diagrams. Carbonized to run in both the Classic and OS X environments.

Intelligent Chaos, Intelligent Mandelbrot, HyperKaosFabian Lidman, 1998Fabian LidmanApplications for viewing the behavior of actual numbers. Like a programmable calculator, but faster. Intelligent Chaos iterates the logistic function in the real numbers. Intelligent Mandelbrot iterates the quadratic map in the complex numbers. HyperKaos is a discontinued HyperCard version of Intelligent Chaos.

OrbitStephen Eubank, University of Texas, Austin, 1986orbit.sitInvestigate a variety of well-known one dimensional oscillators and two dimensional discrete maps. Requires a lot of pampering to get it to work on a PPC machine. (Note the copyright date.)

Quadratic MapT. Abbott, N. B. Tufillaro, J. P. Reilly, 1993quadratic-map.sitDraws time-series, bifurcation diagrams (in color), and phase space diagrams (which are not very useful as far as I'm concerned). Users can't enter their own functions. There are menu items for adding sound, but they don't do anything on my machine. One of the authors has a page describing the program, but it is no longer available for download. This program and Bouncing Ball (see above) go with the book An Experimental Approach to Nonlinear Dynamics and Chaos, the entire contents of which are online in html or pdf format.

StdMapJames D. Meiss, University of Colorado, 1994-2000James D. MeissStandard map orbits were drawn using this program. Has other features for exploring nine different area preserving maps. A nice piece of software with good documentation. Fun to play with, too. Also available in a version for OS X.

Fractal Dimension

CoastlineBoston University Polymer Center, 1995Exploring Patterns in NatureA great little tutorial that walks you through the conceptual steps behind fractal dimension. Construct a simulated coastline by successively crumpling a line segment. Repeatedly measure its length by covering it with ever smaller boxes or lines. Plot size versus number on a log-log graph. The slope of the line of best fit is an empirical measure of your coastline's fractal dimension. Perfect for teachers looking for a way to demonstrate this procedure to their students. One of nine related simulations from Boston University's Polymer Center. Also available in a version for Windows.

Fractal DimensionBoston University Polymer Center, 1995Exploring Patterns in NatureAn application for determining fractal dimension from pict or MacPaint files (also comes with a few sample images). Accepts color and grayscale images, but does its analysis on a black and white bitmapped rendition of the original. Uses the box counting and circle methods. Does not chose box sizes automatically; which means a lot of pointing, clicking, and watching on the part of the user. Automatically fits both data sets to its own straight line so that the two techniques may be compared. One of nine related simulations from Boston University's Polymer Center. Also available in a version for Windows.

Fractal Dimension CalculatorPaul Bourke, 1993Paul BourkeA box-counting application. Give it a black and white Pict image and it will count the number of boxes needed to cover the image. Automatically runs through different sized boxes but does not analyze its own data, which means you'll need some sort of statistical application to come up with a final number. Fractal dimensions calculated in this book are based on data from this program. The accompanying instruction manual is worth reading for its own sake. Includes a 3D version that I have never tested. (3D stuff is just too hard to learn) Also available in a version for OS X running X-Server and Linux.

Iterated Function Systems

Fractal Attraction, IFS Fractal Movie IKevin D. Lee & Yosef Cohen, Sandpiper Software, 1991fractal-attraction.sitA reader was kind enough to email me a copy. Draws IFS fractals from a template of polygons. This program was used to create IFS Fractal Movie I — a stand alone program that does nothing other than play a short movie showing an evolving IFS called "The Claw". This movie is still available in several software archives even though Fractal Attraction disappeared a long time ago. Apparently it was sold commercially with an 80 page instruction book. Check out the reference at amazon.com.

IFSPaul Bourke, 1989Paul BourkeSo far, this is the easiest IFS program to use. I still find that the images generated look nothing like what I predict, however. Uses two different methods: hopalong (which makes images materialize from a haze of dots) and polygon (which is a bit more intuitive). The author also has a version for generating random IFS patterns.

IFSJames D. Meiss, University of Colorado, 2000James D. MeissAs difficult to learn as all the rest, but much faster and more stable. The interface can be a bit unfriendly. Comes with a slide show feature for the truly lazy.

Koyn Fractal Studio DemoTerry Koyn, 1993koyn-fractal-studio-demo.sitThe interface and instructions are far from intuitive. Comes with a few sample IFS files, but I was unable to make my own. Probably because this is just a demo. The retail version costs an unbelievable $119.95 according to the Read Me file or $149.95 according to an article in Byte magazine. (Order before June 30, 1995 and get it for the "discount" price of $89.95.) The author bought koyn.com in December of 1997 but as of June 2002 a related website still does not exist.

Sierpinski-TriangelnMartin Wiss, 1997sierpinski-triangeln.sitA minimal little program that draws the Sierpinski Triangle. You can play around with the parameters, but that's it. Written in Norwegian, but simple enough that even Americans can figure it out.

Fractal TreesSimon Woodside, 1997Simon Woodside [last url]fractal-trees.sitDraws simple stick figure, fractal trees. The original, Classic version is an exceptionally unfriendly program with no user interface. None! Its parameters can only be modified by editing the source code. The version for OS X is orders of magnitude better, however.

KochToyJim Burgess, 2001Jim BurgessA conceptual, non mathematical program. Rearrange the generator with your mouse and see the effect on the fractal simultaneously. No dialog boxes. No numerical parameters to enter. No programming language to learn. L-Systems reduced to their essence.

L-Systems, 3D L-SystemPaul Bourke, 1991Paul BourkeDraws recursively defined fractals like the Peano monster, Koch coastline, Sierpinski gasket, and many others that look like plants or corals. The L-System Manual is worth reading for its own sake. Also available in a 3D version.

Line FractalGabriel Nivasch, 1999Gabriel NivaschRenders over a dozen different IFS fractals including the usual — Koch coastline, Sierpinski triangle, etc. Has two basic features: run through the iterations step-by-step or zoom in. The second feature is surprisingly fast at times.

LParserNick Porcino, 1998Nick PorcinoI was able to get it to render a few 3D images, but I don't really know how this program works or what it does. LParser is too specialized for a generalist like myself. Based on Lparser by Laurens Lapre.

LSystemsBryan Horling, 1996Bryan HorlingRenders fractals from simple recursion instructions. Colorful and visually appealing. The author has also written a nice, compact paper on branching systems in nature (plants, corals, etc.).

pfg (Plant and Fractal Generator)Przemyslaw Prusinkiewicz, 1988-92University of Calgary [ftp link]Draws fractal images and plant-like branching structures using L-systems with "turtle interpretation". Fast, but difficult to work with. No options. No interface for producing your own parameter sets (must be coded by hand using a text editor). The author has a home page, but the only way to acquire this program is through an obscure ftp site.

Mandelbrot & Julia Sets

Aros FractalsAros Magic Research, 1996Aros Magic ResearchMandelbrot set, Newton's method, and a third fractal type I've never seen before that looks something like a moire pattern. Zoom in, but not out. Animated colors. Unusual interface with limited options. Also available in a Windows version.

Cheap MandelConor McCarthy, 2001Conor McCarthyExamine the Mandelbrot set. A beginner's attempt at programming for the Mac. The version for OS X is not much better.

EasyFractAlessandro Guzzini, 1997easyfract.sitEasy, yes, but there are better programs out there. Zoom into the Mandelbrot set, play with the colors, but that's all. Available in English and Italian versions.

EasyFractalBerkhan Software, 2003Berkhan SoftwareZoom into the Mandelbrot set, check out the corresponding Julia set, play with the colors, try a different function. Comes with a collection of preset parameter files, some of which are interesting and rather unusual. Available in German and English versions for both the Classic OS and OS X.

f_zoomAndreas Warnke, 1996f-zoom.sit"Zoom in die Mandelbrotmenge." A cute little nothing of a program for zooming into the Mandelbrot set. A rare piece of überminimalism from Germany. According to the AltaVista Translator, "Zoom in die Mandelbrotmenge" means "Zoom shot into the almond bread quantity."

Floating FractalsAdam Smith, 1994floating-fractals.sitZoom in and explore 11 different fractals (half of them are variations on the Mandelbrot set). Fast and easy to use.

Fractal (a.k.a. Fractal Generation Program)Cathy & Tom Saxton, 1999Tom & Cathy SaxtonIdle Loop SoftwareA nice simple program. Draw the Mandelbrot set. Zoom in a step at a time. Draw the Julia set associated with a point in the Mandelbrot set. Zoom into that a step at a time. Also available in a version for Windows. (Not to be confused with Fractal Generation by Laurent Thil listed below under Miscellaneous).

Fractal ArtistAlexei Lebedev, Phronesis Software, 1992fractal-artist.sitAnother Julia-Mandelbrot explorer. An obscure program I found in a Japanese archive of the comp.binaries.mac newsgroup.

Fractal Domains (formerly FracPPC)Dennis C. De Mars, 1994-2000Fractal DomainsExplore the Mandelbrot set. Switch to the accompanying Julia set. Rather plain interface with lots of control panels floating around. The version for OS X is much nicer. FracPPC was the original, very compact freeware version.

Fractal ObservatoryMarcio Luis Teixeira, Trilobyte Software, 1990fractal-observatory.sitAnother program that doesn't use the Mac interface. Halftone grays meant to be viewed on a black and white monitor. Looks and behaves like Super MANDELZOOM.

Fractastic!Jake Olevsky, 1998Jake Olevskyjakeo.comA nice piece of fractal eye candy. Draws eighteen different fractals including Hénon attractors, Julia sets, Newton's method, and Mandelbrot sets up to the sixth power and does it fast. Includes dynamic zooming and numerous coloring methods with crazy-fast animation as an option.

JavaQuatGarr Lystad, 2000Garr LystadA Java applet for exploring Mandelbrot sets, Julia sets, and the sets in between. Uses quaternions (complex-complex numbers of four real parameters). Stretch your visualization muscles as you view two-dimensional slices through the four-dimensional mother of all sets. Oversized interface makes it hard to work with on smaller monitors.

Julia O' Matic, newer.JOMJim Burgess, 2001Jim BurgessA split window Julia-Mandelbrot explorer. Click on the Mandelbrot set on the right and see the accompanying Julia set on the left. Then zoom into either set. A minimal program with few options and documentation that's a little hard to find. A version that draws higher order sets, called newer.JOM, appeared one day at xahlee.org and is still there. (Some versions have the apostrophe on the wrong side of the O.)

Julia's DreamReinoud Lamberts, 1991julias-dream.sitMany of the Julia sets in this book were drawn using this program. Generates real-time images of Julia sets as you roam around the complex plane with your cursor. The author's home page from 1996 still exists, but it doesn't mention the program. (Julia's Dream was also the name of a pizza: basil pesto, ricotta and mozzarella, topped with garlic, broccoli, and spinach. $9.99 for a 12" pie. According to a now defunct web page, anyway .)

Julia's NightmareBen Davenport, 1995julias-nightmare.sitA full-color sequel to Julia's Dream. I have nothing else to say about this program.

MandelBrainDanny Brewer, 1995mandelbrain.sitExplore the Mandelbrot set. Comes with more preset color palettes than any other program so far.

MandelBrowser, Mandella, MicroMandellaJesse Jones, 1995mandelbrowser.sitMandelbrot sets, Julia sets, Newton's method, and a few others. Over 50 different sets and numerous color schemes. Fast, PPC native code. Mandella is an older 68k version that also draws strange attractors. MicroMandella is a stripped-down version that requires an FPU. Mandella was once sold with a book (now out of print) called Fractals for the Macintosh published by Waite Group Press (now a part of Macmillan Computer Publishing).

MandelMovieShinichiro Hirama, 2001Shinichiro Hirama [last url]mandelmovie-hirama.sitRoam around the Mandelbrot set until you find a region of interest, then create a QuickTime movie that zooms into that spot. Available in Japanese and English.

MandelMovieMichael Larsen, Dynamic Software, 1995Dynamic SoftwareThe Mandelbrot-Julia hybrid and Julia cascade movies in this book were rendered in 1992 at the University of Wisconsin-Milwaukee, School of Education using this commercial program. Haven't used it since, so I can't comment on it. Dynamic Software claims to offer other chaos and fractal programs, but sending them any money seems risky. Some of my favorite pages on their website include: a calendar of upcoming events dated 1995, an on-line book composed entirely of broken links, and a grossly overpriced $500 program designed when Macs ran on System 7. Crazy backgrounds render some pages unreadable.

MandelNetChristoph Pfisterer, 1996mandelnet.sitDraw Mandelbrot sets using the combined power of several computers on an AppleTalk network. Set up master and slave applications on your network. Link the master to the slaves and away you go. Very simple to set up and use, but completely unnecessary. Can only draw one simple, small Mandelbrot set that does not require massive parallel processing power. Still, it's a pretty neat idea.

MandelZotDavid Platt, Think Technologies, 1998David PlattMost of the Mandelbrot sets and some of the Julia sets in this book were drawn using this program. I have been using it since 1991 so I'm used to it. Apparently, a commercial version called FractalMagic is also available from Sintar Software, but I can't confirm any of this. Comes with extensions for drawing Lyapunov space images that I can't get to work.

MandleAcidDerek Greenberg, The Bone Factory, 1993mandleacid.sit"A video drug for the Macintosh." Eye candy, pure and simple. Way cool, but useless for those interested in mathematics. No theory behind what you're looking at. Version 2.0 is orders of magnitude faster than 1.0, comes with a great wacky interface, and will completely coat even the largest monitor with psychedelic glop. I really like this program, but it should be spelled "MandelAcid".

Parallel Fractal CarbonDean Dauger, 2001Project AppleseedDauger ResearchA program for testing the computing power of a distributed computing network. Generates a preset image of the Mandelbrot set, then tells you the speed of your parallel processor network in megaflops. If you only have one computer, then it tells you the speed of just that computer. A test program with no options. — zero. It runs and it's done. Also available in a version for OS X with more than zero option called AltiVec Fractal Carbon.

PhoenixAdrian Platts. no dateAdrian PlattsDeliberately styled interface with options, options, options. Auto zooms, generates relief maps, makes movies, plays background MIDI and MP3 files, and acts as a screen saver. It also draws several kinds of Mandelbrot and Julia sets. Comes with yet more options that I haven't described. A version for OS X is reportedly in the works.

PowerXplorerAllesandro Levi Montalcini, 1996Alessandro Levi MontalciniBare bones simple. No interface, no documentation, no options. Draw a box around the region you want to explore and let go. PowerExplorer zooms you in. Intended primarily for testing computing speed. Has a bug that replaces the menu bar with a blank region in full screen mode. Also available in a Japanese version.

RatioJuliaChristoph Stroh, 1998Christoph StrohDraws Julia sets of rational functions. A unique program that is difficult to use. Has a combination command-line, menu-driven interface. Not much fun.

Super MANDELZOOMRobert P. Manufo, 1988super-mandelzoom.sitAn antique. Meant to be viewed on a Mac SE or thereabouts. Uses halftones instead of grays. It's been a long time, but the author is still out there working on fractals and whatnot.

XaoS Fractal ZoomerJan Hubicka, 1997xaos.theory.orgMandelbrot sets of powers 2-6. Zoom in dynamically, then switch to the Julia set at the same level of magnification and the same point. Nice for showing the quasi-self-similarity between the two sets. Other fractals included: octal, Newton's method, Barnsley, and Phoenix. Comes with a really neat option that remaps the complex plane. Very fast, optimized code that doesn't try to redraw the entire window. Intentionally ugly interface takes some getting used to, however. Available for nearly every platform: Amiga, BeOS, DOS, Linux, Mac OS, OS/2, and even Windows (or so they claim).

Music

BifurcationOscillator, Logistic SynthJae Ho Chang, 1996.bifurcationoscillator.sitOrbit diagram and chaotic music generator. Watch the evolution of an orbit and then transform it into a music-like series of notes saved in AIFF format. Logistic Synth is an equivalent program for generating sounds in real time.

ChaoticPianolaThe Boltzmann Toy Factory (Lars Rosenberg), 1998Boltzmann Toy FactoryListen to the behavior of the iterated logistic function played on the QuickTime Roland digital piano. A different approach to the exploration of a dynamical system. Sounds like Philip Glass when periodic and Schönberg when chaotic. Since there is a sort of structure to the chaotic regime, ChaoticPianola is briefly listenable from time to time. Also plays a random function for comparison or to reseed a stuck or stale pattern.

Newton's Method

EscapeFractalsYves Meynard, 1996Yves MeynardThis program draws fractals using an escape-time method developed by Clifford Pickover. Visually appealing and fast. Not to be confused with EscapePPC (filed above under Mandelbrot & Julia Sets).

Halley MapsYves Meynard, 1997Yves MeynardThis program produces interesting looking fractals using Halley's root approximation method for equations up to the twelfth degree. Windows must all be square and less than 512 pixels in size. No animated colors or dynamic zooming and yet I still like it.

NewtonFractalStefan Messmer, 2001Stefan MessmerThe latest Classic version of NewtonFractal works, but just barely. This is a very slow application. Also available for OS X (see above), Windows, and Java.

Terrain & Landscape Modeling

Fractal ContoursJim Cathey, 1985. Eric Traut, 1995fractal-contours.sitftp.mactech.com [ftp link]A wire frame fractal terrain generator based on algorithms developed by Lucas Film. I have never been able to run the original program without having it crash immediately. Looking at the icon I thought someone was playing a joke on me. Then I came across a MacTech article called "Extreme Advantage" describing how Fractal Contours could be completely overhauled to run on a PPC. The author has made all eight reincarnations with their source code available to the general public. I bet this thing was pretty advanced in 1985 when the Mac was still in its infancy, but it doesn't look like anything special now.

Fractal IslandsScott Berfield, Parity Productions, 1985fractal-islands.sitAnother relic from 1985. Guess what? It crashes after every run on a PPC Mac.

Fractal!Ed Rotberg, Gonzo Systems, 1992-93Ed RotbergDraws interesting snowcapped mountains with blue lakes in the foreground. Play with lighting and color schemes. Use it to render a desktop background if you wish.

Matt's FractMatson Dawson, 1995matts-fract.sitFly around over a fractal-generated terrain. Not detailed or realistic in any way.

Miscellaneous

CarpetKevin Whitley, Think Technologies, 1987-88carpet.sitA simple program for stamping carpet or gasket fractals. Ancient, but it still runs. Comes with a severely damaged tutorial file, however.

ChaosGame, SnowCrystalK. Hirai, 1997K. HiraiTwo small programs that use some sort of recursive action to generate what appear to be fractal images. The user interface is a set of chords of varying thickness on a pair of circles: x and y. (SnowCrystal has three circles: i, j, and k.) Moving the chords around the circle transforms the image. Anything beyond that is a mystery to me as the user manual is written in Japanese.

Chaos Under Control Brian S. Macherone, 1994Fractal Geometry Summer WorkshopFound this obscure suite of applications in the class notes for an August 2000 workshop at Yale University, taught by, of all people, Benoit Mandelbrot (the father of fractals) and Michael Frame (coauthor of Chaos Under Control, a book on chaos, fractals, and cellular automata for nonscientists). Included with the notes was the software to accompany Frame's book. All five applications are creaky and slow, which obscures their utility as learning tools. They are also a bit hard to appreciate when removed from the context of this one particular class.

Cubic Oscillator ExplorerBruno Degazio, 1995University of StuttgartCan't download it for review. "Communications exception (-244)." Would someone out there send me a copy?

Escape (a.k.a. EscapePPC)Graham Anderson, 2004EscapePPCA comprehensive application for exploring over 20 different types of fractal images and strange attractors. Unregistered versions can only render Mandelbrot sets, however. Many, many features to explore for those with a serious interest. Carbonized to run in both the Classic and OS X environments. (Not to be confused with EscapeFractals: a Newton's Method application for the Classic OS.)

FractalPaul Bourke. No datePaul BourkeThis program renders a diverse array of fractal objects: the Mandelbrot set, Julia sets, Newton's method, diffusion patterns, Hénon orbits, several strange attractors, plus a few more. Does a good job drawing strange attractors. (Allows you to play around with the parameter values and see what changes.) Does a poor job drawing the Mandelbrot set. (It's slow and ugly.) Has a strange quirk to it that I found annoying: accessing the menu bar wipes out the display's memory and starts the rendering process over from the beginning.

Fractal ExplorerPeter Stone, 1999Peter StoneEasily the largest collection of preset fractals: Mandelbrot sets, Julia sets, escape fractals, Newton's method, quaternions, and others. Enter your own complex functions and see what happens. Fairly fast, but the interface is too clumsy for this to be fun. Zooming in is especially tiresome. There were only 74 visitors to the program's home page from March 1999 to February 2000. Something doesn't seem right. Won't you please visit this lonely site and cheer it up?

Fractal Lab KitRonald T. Kneusel, 1994fractal-lab-kit.sitA command-line-driven IFS program. Why bother programming for the Mac if you're not going to make use of the GUI interface? The full package also includes one program for Julia set exploration and another for the Mandelbrot set. I can only get them to draw one image each before they crash.

Fractal Studio (formerly Fractals)Keiron Liddle, Aftex Software, 1999Aftex SoftwareDraws 30 different fractals. Has a ton of different options to play with. I haven't had the energy to fully explore this monster.

FractalDesignerMartin Hairer, 2000HairerSoftRenders preset as well as user-defined functions. CodeWarrior users can even program it to draw Markus-Lyapunov fractals, but others have to live with the presets. I appreciated the flexibility, but was disappointed by the lack of speed and mediocre user interface.

Fractal GenerationLaurent Thil, 2001Laurent ThilDraws Mandelbrot, Julia, and "Magnetisme" fractals in your choice of colors and rendering schemes in a teeny tiny little window. Also renders disorienting QuickTime movies that make you feel like an ant crawling over a fractal textbook. Comes with preset parameter files to get you started — an absolute necessity given that the program is basically a "black box". The author is probably the only person who really knows how to use it. (Not to be confused with Fractal Generation Program by Cathy and Tom Saxton filed above under Mandelbrot & Julia Sets.)

Gerry's Mandelbrot SetGerry Beggs, 1998Gerry BeggsMore than just the Mandelbrot set, it also creates Julia sets, IFS images, a fractal tree, the Lorenz attractor, bifurcation diagrams, Newton's method, and Martin's mapping (Hopalong). Gives you the ability to check out a lot of different fractals using just one application. Also available in a version for OS X.

IconMakerJim Burgess, 2001Jim BurgessPlots points in the orbit of four unique iterative mappings up to the twelfth power. Color indicates the density of points at a particular location. Sliders permit you to change all five of the parameters that control the mapping. You can also bookmark interesting combinations and return to them later. This is a necessary feature given that it is quite easy to get lost among all the numbers.

Iterative FunctionsHarold Brochmann, 1992iterative-functions.sitAn odd suite of six programs: diffusion limited aggregation, real orbits, complex orbits, the Mandelbrot set, logistic equation, and population dynamics. Lousy interface that makes no use of Mac features. A time capsule from the bad old days. Great in terms of content, however. Someone should rewrite this suite and wrap it in an up-to-date interface.

LyapunovAndrea Pellizzon, 1994lyapunov.sitLyapunov diagrams in this book were drawn using this program. It is the only program I know dedicated to drawing this kind of fractal. The author has a home page, but it never mentions this program.

MacFractintTim Gilman, 1999Fractint for the MacintoshRoom for one more. Won't you join us? A well loved Windows-Intel classic, Fractint may finally make it to the Mac OS. The current beta version has a lot of preset fractals to play with, but the program as a whole is obviously still under construction.

Persian RugPaul Cantrell, 1996persian-rug.sitPersian Rug is a control panel that generates random fractal patterns for your desktop. Don't like your current desktop? Open the control panel and click the "change" button. A really great idea, but unfortunately I was never able to run it for very long without having my machine freeze.

Stone's Free Mac ProgramsIshihama Yoshiaki, 1996-2001Stone's Free Mac ProgramsA prolific programmer, Stone has written around a hundred freeware applications for the Mac OS including about a dozen chaos and fractal programs. All are quite primitive and unpolished.

Fractals BundleAlessandro Levi Montalcini, 1992fractals-bundle.sitA suite of programs designed to run on 68k Macs with a math coprocessor. The After Dark module (called "Fractal" in the last version, but known as "Mandelbrot" in earlier versions) draws regions of the Mandelbrot set, which it then saves as Pict files. Comes with a program for viewing the picts and a mover based on the old Font/DA mover. As the author seems to admit, such a system is of little use now. There is no mention of this bundle at his website.

HyperCard Stacks

With the ascent of the World Wide Web there really doesn't seem to be much of a need for HyperCard anymore. It was a great application for its time, but I doubt I will add anything new to this category.

Chaos TheoryMike Atanasio, Wild Card Software. No datechaos-theory.sitHyperCard stack. Listen to the time-series of the logistic equation. A one card stack. No date, but it appears that it was created in 1990.

HyperCard ChaosA.J. Roberts, University of Adelaide, 1991hypercard-chaos.sitThe most technical of the HyperCard stacks. A tutorial in dynamical systems with applications and interactive demos. Serious stuff, but doesn't always display well. The interactive demos are a bit erratic. I loved many of them, but a few were complete mysteries.

Mandelbrot Fractal CompanionEvan T. Yeager, 1993mandelbrotfractalcompanion.sitA brief HyperCard tutorial of complex numbers, fractals, the Mandelbrot set, and how to construct a computer program. In living color (something rare in HyperCard).

MandelMusicRoger R. Espinosa, Donna Iadipaolo, Jim Brunberg. No datemandelmusic.sitHyperCard stack. They call it a chaotic music generator, but I can't confirm this. No mathematical description of what it actually does. Your choice of 12 different instruments.

Floating Point Unit Required

A floating point unit (FPU) is a coprocessor that handles intensive mathematical calculations to relieve some of the burden on a computer's central processor unit (CPU). Applications designed to access an FPU's subroutines will run faster than those that don't; that is, if your computer has one. FPUs are currently extinct and have been so since about 1995. Computers are no longer designed or manufactured with these units. If you try to run an FPU program on a machine without one it will likely crash. If you're lucky you might just get an error message. There is a workaround for those who insist on trying things for themselves. I've managed to execute almost all of these applications on a 603 PPC Mac using a control panel called SoftwareFPU, but with only mixed results. Some programs run awkwardly under the emulator. Some run just fine. I have not tried this on a G3 or newer machine nor will I ever. There are so many non-FPU programs out there with similar capabilities that these few dinosaurs won't be missed. I present them here in an effort to maintain a complete archive.

ab FractalEden Software, L. Pieniazek, 1994ab-fractal.sitI don't have time for this. It draws the Mandelbrot set, Julia sets, and maybe some other sets, but is intolerably slow. I couldn't believe that anyone would ask a shareware fee for this. Requires an FPU.

Animated Fractal GeneratorJerry Goldstein, 1990animated-fractal-generator.sitOne of the slowest programs around. I estimate that a 300 MHz PPC machine would take an hour to complete one 640 × 480 image. I can't confirm this as I didn't feel like waiting. According to the manual, this program can render Mandelbrot, Julia, and "other specialized varieties of fractals." Requires an FPU.

ARTAbrotGeorge Warner, 1994artabrot.sitDoes not require an FPU but does require a DSP3210, which means it will only run on a Quadra 840AV or Centris 660AV — the two machines that used the AT&T 3210 Digital Signal Processor to run the ARTA (whatever that is). I cannot review this program since I do not have either of those machines.

Fractal WizardThomas Okken, 1992fractal-wizard.sitMandelbrot sets, Julia sets, Newton's method, IFS, and a few others. Quite a range of different sets, but it's basically another slow antique. Has a nice pull down menus briefly explaining the mathematics. Requires an FPU.

KaosReinoud Lamberts, 1990kaos.sitKaos produces IFS images that are quite unique, but it is painfully slow. After running half an hour on a 300 MHz 603e PPC, I had a smudge that occupied one-third of the screen. An interesting looking smudge, but it wasn't worth the wait. The author's home page from 1996 still exists, but it doesn't mention the program (or much of anything else).

MacMandelMovieMakerpad-parrot productions, J. Geagan, 1993macmandelmoviemaker.sitCreates a sequence of PICT files spiraling into a point on the Mandelbrot set. In only 28 hours I was able to generate a ten second movie of a 101 frames. It was a truly stunning accomplishment [read sentence dripping with sarcasm]. Requires an FPU. Geagan is also responsible for the Mac shareware classic 99 Bottles — an application that sings the famous time-killing, travel song for you. Set it at a thousand bottles and pretend your in the audience at an Andy Kaufman performance. Set it to a million bottles and have it sing along with MacMandelMovieMaker while it generates a half hour video for you. On second thought, a million bottles may not be enough. 99 Bottles does not require an FPU.

MandelScope, MandelMakerAnthony S. Ku, 1993mandelscope.sitExplore the Mandelbrot set with MandelScope and then draw it using MandelMaker. Why not combine these features into one program? MandelScope has an unusual interface that might have seemed clever at the time, but looks ridiculous now. MandelMaker uses a divide and conquer algorithm that should speed things up, but still runs slowly. Doesn't use the symmetry of the Mandelbrot set across the real axis. Requires an FPU.

MandelTVEd Ludwig & Ken Abbott, Abbott Systems, 1990mandeltv.sitA relic from the days of System 6. MandelTV is a desk accessory for exploring the Mandelbrot set. Abbott Systems also sells a commercial fractal software package called Kaleidoscope that I have not reviewed. It also requires an FPU!

The first chapter introduces the basics of one-dimensional iterated maps. Say what? Take a function y = ƒ(x). Substitute some number into it. Take the answer and run it through the function again. Keep doing this forever. This is called iteration. The numbers generated exhibit three types of behavior: steady-state, periodic, and chaotic. In the 1970s, a whole new branch of mathematics arose from the simple experiments described in this chapter.

The second chapter extends the idea of an iterated map into two dimensions, three dimensions, and complex numbers. This leads to the creation of mathematical monsters called fractals. A fractal is a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry. They are quite interesting to look at and have captured a lot of attention. This chapter describes the methods for constructing some of them.

The third chapter deals with some of the definitions and applications of the word dimension. A fractal is an object with a fractional dimension. Well, not exactly, but close enough for now. What does this mean? The answer lies in the many definitions of dimension.

The fourth chapter compares linear and non-linear dynamics. The harmonic oscillator is a continuous, first-order, differential equation used to model physical systems. The logistic equation is a discrete, second-order, difference equation used to model animal populations. So similar and yet so alike. The harmonic oscillator is quite well behaved. The paramenters of the system determine what it does. The logistic equation is unruly. It jumps from order to chaos without warning. A parameter that discriminates among these behaviors would enable us to measure chaos.